For the purposes of this question, a homology theory is a covariant functor from the homotopy category of finite pointed CW complexes to graded abelian groups, and a collection of connecting homomorphisms, which satisfy the Eilenberg-Steenrod axioms for reduced homology, minus the dimension axiom. So this amounts to the statement that for $f:A \to X$, the connecting homomorphism and the functor grades give a long exact sequence using the cone of $f$, along with excision (where relative homology is just homology of the cone); I don't think(?) I need the suspension axiom. Dually, a cohomology theory is a contravariant functor and connecting homomorphisms satisfying the dual axioms.
The Eilenberg-Steenrod theorem, as I understand it (EDIT According to Qiaochu's second comment below, though, I understand incorrectly!), says that if two homology theories have isomorphic values on $S^0$, then this isomorphism extends uniquely to a natural isomorphism between their underlying functors, and their connecting homomorphisms agree under this isomorphism (actually this can't quite be true -- you could always replace the connecting homomorphism with its negative. I'm not sure how far from the truth this is, though.) Similarly for cohomology.
How about the converse -- given a graded abelian group $A$, does there exist a (co)homology theory which takes $S^0$ to $A$?
Brown representability isn't what I want -- that requires you to already have the functor and builds a spectrum (in fact it's almost the opposite, and it's suspicious that I seem to be replacing the role of the spectrum with a lowly graded abelian group).
It does seem to be related to the Atiyah-Hirzebruch spectral sequence. I don't imagine the spectral sequence is well-defined if I haven't actually specified a functor, but if it were, then I could attempt to piece together the functor from the output of the spectral sequence. The difficulty of doing this makes me suspect that I've got it all wrong.
My secret goal here is that I'd like a purely formal construction of ordinary homology. The most formal way I know to do this is to define it representably using an Eilenberg-Mac Lane spectrum. But it still takes some work to construct this spectrum, and I'd like to circumvent this by appealing to more general machinery.
EDIT In light of Qiaochu's answer below, there's an important lesson to be learned. The Eilenberg-Steenrod axioms including the dimension axiom are enough to calculate the ordinary (co)homology of any finite CW complex. Removing the dimension axiom gives the right notion of generalized (co)homology theory. Nevertheless, replacing the dimension axiom in the obvious way by specifying a different value for the (co)homology of $S^0$ is not enough information to determine a (co)homology functor, so the Eilenberg-Steenrod axioms + an alternate dimension axiom will not provide enough calculational rules to determine the generalized cohomology of an arbitrary CW complex.
The fact that ordinary (co)homology is determined qua (co)homology theory by its value at a point, then, is a special fact which resembles the special fact that a $K(A,n)$ is determined qua space by its homotopy groups. I wonder if there is a connection...